Debye Length And Plasma Skin Depth: Two Length Scales Of .

J. Plasma Phys. (2017), vol. 83, 595830106doi:10.1017/S0022377817000022c Cambridge University Press 20171Debye length and plasma skin depth: two lengthscales of interest in the creation and diagnosis oflaboratory pair plasmasE. V. Stenson1, †, J. Horn-Stanja1 , M. R. Stoneking2 and T. Sunn Pedersen1,31 Max Planck Institute for Plasma Physics, 17491 Greifswald & 85748 Garching, Germany2 Lawrence University, Appleton, WI 54911, USA3 University of Greifswald, 17489 Greifswald, Germany(Received 15 September 2016; revised 27 December 2016; accepted 4 January 2017)In traditional electron/ion laboratory plasmas, the system size L is much larger thanboth the plasma skin depth ls and the Debye length λD . In current and planned effortsto create electron/positron plasmas in the laboratory, this is not necessarily the case. Alow-temperature, low-density system may have λD L ls ; a high-density, thermallyrelativistic system may have ls L λD . Here we consider the question of whatplasma physics phenomena are accessible (and/or diagnostically exploitable) in thesedifferent regimes and how this depends on magnetization. While particularly relevantto ongoing pair plasma creation experiments, the transition from single-particlebehaviour to collective, ‘plasma’ effects – and how the criterion for that thresholdis different for different phenomena – is an important but often neglected topic inelectron/ion systems as well.Key words: astrophysical plasmas, magnetized plasmas1. IntroductionThe large mass imbalance between ions and electrons – and the resulting separationof the two types of particles’ length and time scales – is a cornerstone of the physicsof traditional plasmas (e.g. Chen 1984; Bellan 2006). There are ‘fast’ phenomenathat involve electron oscillation (with the ions stationary) and ‘slow’ phenomena thatinvolve ion oscillation (with the electrons reaching equilibrium ‘instantly’ – i.e. onmuch faster time scales than the motion being considered). In the governing equations,terms with the ratio me /mi (electron mass divided by the ion mass) are frequentlydiscarded. Plasma mass density and centre-of-mass velocity are approximated by theion mass density and centre-of-mass velocity. Intraspecies temperature equilibration isimpeded, because, unlike a collision between two equal mass particles (which maytransfer up to 100 per cent of one particle’s energy to the other), a collision betweenparticles with a large mass difference transfers comparatively little energy. These arejust a few basic examples. More complex plasma phenomena are in turn built on thefoundations of mass asymmetry.† Email address for correspondence: evs@ipp.mpg.deDownloaded from https:/www.cambridge.org/core. Balfour Library (Pitt Rivers Museum), on 19 May 2017 at 07:41:09, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377817000022

2E. V. Stenson, J. Horn-Stanja, M. R. Stoneking and T. Sunn PedersenTherefore, the concept of a ‘pair plasma’, comprising particles with oppositecharge but equal mass, requires all of plasma physics to be revisited from theground up. Sometimes the end result of the rederivation is no more than an extrafactor of 2, as a term that was once negligible compared to its neighbour is nowequally important. If, on the other hand, those two terms have opposite signs, theynow cancel, fundamentally changing the result. Behaviour can go from linear tononlinear, be dominated by a different aspect of the physics or disappear entirely.Hundreds of papers have been written on the topic, employing a variety of differenttheoretical treatments – relativistic and non-relativistic; kinetic and multi-fluid; linearand nonlinear (see, for example, Tsytovich & Wharton 1978; Stewart & Laing1992; Iwamoto 1993; Berezhiani & Mahajan 1994; Blackman & Field 1994; Zank& Greaves 1995; Verheest & Lakhina 1996; Mahmood, Mushtaq & Saleem 2003;Bessho & Bhattacharjee 2005; Gary & Karimabadi 2009; Lopez et al. 2012; Helander2014; Liu et al. 2015; Edwards, Fisch & Mikhailova 2016, just to name a small butdiverse selection).Naturally, interest in creating a pair plasma in the laboratory goes as far back as thefirst theoretical conceptions of such a thing (Tsytovich & Wharton 1978). However,this is a significant experimental challenge. It is also one that different researchersaround the world approach in a variety of different ways. Progress to date can besummarized as follows:Pure positron plasma electron beam. Two-stream instability seen in a chargeneutral system. Electron Debye length exceeded the electron beam diameter (λDe de )(Greaves & Surko 1995).Laser-driven, relativistic positron/electron beams. Charge neutrality approachedasymptotically, but effectively achieved. Plasma skin depth of the order of or slightlysmaller than the beam diameter (ls d); simulations predict such a system will exhibitsome collective behaviour (Wilks et al. 2005; Chen et al. 2015; Liang et al. 2015;Sarri et al. 2015).Carbon fullerene pair plasmas. Many Debye lengths achieved. Electrostatic modesinvestigated. Gyroradius plasma radius (Oohara & Hatakeyama 2003; Oohara, Date& Hatakeyama 2005; Kono, Vranjes & Batool 2014).Low-temperature electrons/positrons in a dipole magnetic field. Only single-speciesexperiments to date (i.e. highly non-neutral). Many Debye lengths achieved withelectrons but not yet with positrons. Target is ten Debye lengths for both speciesin the same system (Saitoh et al. 2010, 2015; Pedersen et al. 2012; Stenson et al.2015).As may be gathered from the above overview, different experimental groups havetended to focus on one of two different parameters as their figure of merit (typicallypreferring the smaller of the two): plasma skin depth ls or Debye length λD . Whilethe Debye length is often used in textbooks as part of the definition of a plasma, itis also true that traditional laboratory plasmas have very small plasma skin depths, afeature that cannot be taken for granted in very low-density systems.In this paper, we review the physical relevance of each parameter. We discusshow small Debye length does not guarantee small plasma skin depth for plasmasthat are thermally non-relativistic, while Debye length can equal or modestly exceedplasma skin depth for plasmas that are thermally relativistic. We note that certaincollective interactions can occur before either multiple plasma skin depths or multipleDebye lengths are reached, and we consider the topic of experimental observablesDownloaded from https:/www.cambridge.org/core. Balfour Library (Pitt Rivers Museum), on 19 May 2017 at 07:41:09, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377817000022

Debye length and plasma skin depth3for different regimes. While particularly relevant to ongoing pair plasma creationexperiments, the transition from single-particle behaviour to collective, ‘plasma’effects – and how that threshold depends on the phenomena of interest – appliesto electron/ion systems as well, but it is often given only a cursory treatment inplasma physics texts and courses. Therefore we will keep the discussion as generalas possible, addressing plasmas both with and without a large mass asymmetry andnoting differences between the two as they come up.2. Plasma skin depth and plasma frequencyThe plasma skin depth is the depth in a collisionless plasma to which low-frequencyelectromagnetic radiation can penetrate (as defined by attenuation of the waveamplitude by a factor of 1/e). Some representative values are listed in table 1.In a traditional plasma, the expression for plasma skin depth is given by ls c/ωpe ,where c is the speed of light in vacuum. The electron plasma frequency ωpe is thecharacteristic frequency for oscillations of the electron density in the ‘coldp electron’limit (i.e. neglecting the effects of thermal motion); it is given by ωpe ne e2 /( 0 me ),where ne is the number density of the electrons, e is the elementary charge, 0 is thepermittivity of free space and me is the electron mass. For electron/positron pair plasmas, a factor of2 appears, because positronsrespond as quickly to density perturbations as electrons do; the picture of a displacedpopulations of electrons sloshing back and forth past approximately stationary ionsdoes not apply. One option is to redefine ωpe , replacing me by me /2; another isto leave the definition the same and modify the dispersion relations accordingly(keeping in mind that the definition no longer has the same physical meaning). Ahybrid approach is often used in the literature: the definition of ωpe is unchanged, butthe term ‘plasma frequency’ is used for 2ωpe . We will use this approach as well,making a point to distinguish between the ‘electron plasma frequency’ ωpe and the‘plasma frequency’ ωp that represents a fundamental frequency of the system and isgiven bys2ne e2 ωp 2ωpe .(2.1) 0 meThis yields the expression for pair plasma skin depthls cc .ωp2ωpe(2.2)In traditional, electron/ion plasmas, the plasma frequency appears in the dispersionrelation for all elementary plasma waves in which electrons are oscillating, and it is A note on the ‘plasma skin depth’ versus the ‘skin depth’ associated with the ‘skin effect’ in ohmicconductors: metals and semiconductors also have a plasma skin depth corresponding to the natural oscillationfrequency of density perturbations to the free electron gas; it has the same form as (2.2), except with mereplaced by the electrons’ effective mass m* to take into account the effect of the ions’ periodic potential.(The quantum of these oscillations, the plasmon, is a big deal in condensed matter physics.) The cutofffrequency is typically in the ultraviolet (UV) (since n 1029 m 3 ), which explains why metals reflect lightin the visible range. At low frequencies, on the other hand (ω ν, where ν is the interspecies collisionfrequency), the attenuation of vacuum electromagnetic (EM) waves is dominated not by collective oscillationsof the free electron gas but rather by ohmic dissipation, which produces the frequency-dependent skin depthfamiliar to those who work with alternating currents (AC). A (collisional) plasma will behave similarly inthat limit. For a rigorous examination, see Fitzpatrick (2008).Downloaded from https:/www.cambridge.org/core. Balfour Library (Pitt Rivers Museum), on 19 May 2017 at 07:41:09, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377817000022

4E. V. Stenson, J. Horn-Stanja, M. R. Stoneking and T. Sunn Pedersenne (m 3 )12101012 (e /e plasma)10201021 –1022 (e /e plasma)10221025lsλcutoff5.3 m3.8 m0.53 mm0.1–0.4 mm53 µm1.7 µm33 m24 m3.3 mm0.6–2.6 mm334 µm10.6 µmCommentsBetween AM and FM radio—MicrowavesWith relativistic correctionaCO2 laserTABLE 1. Plasma skin depths (ls ) for a range of electron densities (ne ), as well as thecutoff wavelength for transmission of incident EM waves (λcutoff 2π ls ), adapted fromAttwood (2009). Except where noted, a non-relativistic electron/ion plasma is assumed.aAs given by Liang et al. (2015) and Sarri et al. (2015).22 2a cutoff for Langmuir waves, which have the dispersion relation ω2 ωpe 3vTek , aswell as for transverse electromagnetic waves (in unmagnetized plasma) and ordinary2waves (in magnetized plasma), both of which have the dispersion relation ω2 ωpe 2 2c k . (The frequency and wavenumber for an oscillating mode of the plasma are ωand k, and vTe κTe /me is the electron thermal velocity, where κ is Boltzmann’sconstant and Te is the electron temperature.) For example, light with ω ωpe will befully reflected while light with ω ωpe will be transmitted.In electron/positron pair plasmas,for the reasons outlined previously, the cutoff for Langmuir and

plasma physics texts and courses. Therefore we will keep the discussion as general as possible, addressing plasmas both with and without a large mass asymmetry and noting differences between the two as they come up. 2.Plasma skin depth and plasma frequency The plasma skin depth is the depth in a col